Book Review: Model theory

Abstract

In 1972 V. I. Lomonosov discovered a technique which settled the longstanding problem of whether or not two commutative compact operators have a common invariant subspace. He actually proved more: If A is a compact operator, then A shares a common invariant subspace with every operator that commutes with it. In fact as he asserted, a slight modification of his proof shows that the conclusion holds if A merely commutes with a compact operator. His technique, which utilizes the Schauder fixed point theorem, was immediately seized upon by many people and used to produce even stronger invariant subspace theorems. The paper A survey of the Lomonosov technique in the theory of invariant subspaces, by C. Pearcy and Allen L. Shields takes us through Lomonosov's contribution to some of its consequences and also discusses the current interesting state of the invariant subspace problem. A goodly portion of this material has also appeared in the monograph Invariant subspaces of H. Radjavi and P. Rosenthal, Springer-Verlag, Berlin, 1973. There is much of interest in this book. The writing is generally brisk and meets a high standard for mathematical exposition. These essays can contribute a great deal to showing students some of the areas of operator theory that have been and are still the subject of considerable research.